{
  "id": "2409.18621",
  "version": "v1",
  "published": "2024-09-27T10:45:02.000Z",
  "updated": "2024-09-27T10:45:02.000Z",
  "title": "A New Bound on the Cumulant Generating Function of Dirichlet Processes",
  "authors": [
    "Pierre Perrault",
    "Denis Belomestny",
    "Pierre Ménard",
    "Éric Moulines",
    "Alexey Naumov",
    "Daniil Tiapkin",
    "Michal Valko"
  ],
  "categories": [
    "math.PR",
    "cs.IT",
    "math.IT"
  ],
  "abstract": "In this paper, we introduce a novel approach for bounding the cumulant generating function (CGF) of a Dirichlet process (DP) $X \\sim \\text{DP}(\\alpha \\nu_0)$, using superadditivity. In particular, our key technical contribution is the demonstration of the superadditivity of $\\alpha \\mapsto \\log \\mathbb{E}_{X \\sim \\text{DP}(\\alpha \\nu_0)}[\\exp( \\mathbb{E}_X[\\alpha f])]$, where $\\mathbb{E}_X[f] = \\int f dX$. This result, combined with Fekete's lemma and Varadhan's integral lemma, converts the known asymptotic large deviation principle into a practical upper bound on the CGF $ \\log\\mathbb{E}_{X\\sim \\text{DP}(\\alpha\\nu_0)}{\\exp(\\mathbb{E}_{X}{[f]})} $ for any $\\alpha > 0$. The bound is given by the convex conjugate of the scaled reversed Kullback-Leibler divergence $\\alpha\\mathrm{KL}(\\nu_0\\Vert \\cdot)$. This new bound provides particularly effective confidence regions for sums of independent DPs, making it applicable across various fields.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-09-27T10:45:02.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "cumulant generating function",
      "dirichlet process",
      "asymptotic large deviation principle",
      "varadhans integral lemma",
      "feketes lemma"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}